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I'm studying Functional Analysis by myself. the following is an exercise while I'm not sure about my answer.

If $X$ is a locally convex space (LCS), show that $X$ is metrizable if and only if $X$ is first countable. Is this equivalent to saying that $\{0\}$ is a $G_\delta$ set?

My proof:

First, suppose an LCS $X$ is metrizable. Then its topology is determined by a countable family of seminorms $\{p_n\}$. put $V_{n,m}=\{x\in X ; p_n(x)<\frac{1}{m}\}$ for every n,m>0. then the family $\{V_{n,m}; n,m>0\}$ is a local basis at 0 which show that $X$ is first countable.

Conversely, suppose an LCS $X$ is first countable, and let $\{V_n; n>0\}$ be a local base at 0, where every $V_n$ is convex and balanced. corresponding every $V_n$, there is a unique continuous seminorm $p_n$. now define pseudo-metric $d(x,y):= \sum 2^{-n}\frac{p_n(x-y)}{1+p_n(x-y)}$. $d$ is a metric and $X$ is metrizable.

For the rest of exercise, I do not have any idea. Please correct my answer. Thanks in advance.

Matt Rosenzweig
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niki
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  • You need to show that $d$ indeed induces the given topology. That is routine, yet must be done. The "rest of the exercise" is the question whether first countability is the same as ${0}$ being a $G_\delta$ set? [It's not, look at some non-metrisable HLCS that you know to find a counterexample.] – Daniel Fischer Jul 02 '14 at 14:39
  • Daniel Fischer: Thanks so much – niki Jul 02 '14 at 16:40
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    @niki: For your information, a metrizable topological space $ (X,\tau) $ is always first-countable. Let $ d: X \times X \to \mathbb{R}{\geq 0} $ be a metric on $ X $ that is compatible with $ \tau $. Then for each $ x \in X $, the collection $$ \left{ \left{ y \in X ~ \middle| ~ d(x,y) < \frac{1}{n} \right} \right}{n \in \mathbb{N}} $$ is a countable $ \tau $-neighborhood base of $ x $. The harder part is to show that a first-countable locally convex topological vector space is metrizable, which you have done above. :) – Berrick Caleb Fillmore Mar 30 '15 at 18:18
  • @BerrickFillmore: thanks for your remark. – niki Mar 30 '15 at 20:33

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